Cosmic ray-modified shocks: appearance of an isothermal jump

We point out that for sufficiently strong shocks, with Mach number $M_1 > \sqrt{\frac{3 \gamma -1}{(3-\gamma ) \gamma }}= 1.34$ ($\gamma=5/3$), the solutions for cosmic ray-modified shocks experiences a bifurcation. As a result, for super-critical flows an isothermal jump forms (which is not a shock). The isothermal jump forms due to the energy diffusion of fast, but energetically subdominant cosmic rays. For super-critical flows the isothermal jump appears regardless of a particular feed-back mechanism from the CRs. The compression ratio at the isothermal jump is $2/ (\gamma-1)=3$, so that in the test particle regime the expected spectrum of low energy CRs experiencing first-order Fermi process is $p = 2\gamma/(3- \gamma) = 5/2$, steeper than conventional $p=2$.